Find the derivative of `y = e^sin x`.

Answer

To find the derivative of the function \( y = e^{\sin x} \), we can follow these steps:

### Step 1: Identify the function
The given function is:
\[ y = e^{\sin x} \]

### Step 2: Use the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative is given by:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
where \( u = g(x) \).

### Step 3: Set \( u \)
Let:
\[ u = \sin x \]
Then, we can rewrite the function as:
\[ y = e^u \]

### Step 4: Differentiate \( y \) with respect to \( u \)
Now, differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = e^u \]

### Step 5: Differentiate \( u \) with respect to \( x \)
Next, differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \cos x \]

### Step 6: Apply the chain rule
Now, substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^u \cdot \cos x \]

### Step 7: Substitute back for \( u \)
Finally, substitute \( u = \sin x \) back into the equation:
\[ \frac{dy}{dx} = e^{\sin x} \cdot \cos x \]

### Final Answer
Thus, the derivative of \( y = e^{\sin x} \) is:
\[ \frac{dy}{dx} = \cos x \cdot e^{\sin x} \]

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